Stop the presses! I agree with Dawn
Also, you said you watch this little girl every day after school for free (you didn't say WHY you watch her for free, or for how long - 30 minutes? 3 hours?) plus she and your daughter are in the same class every day together. Then you mentioned something about "her other daycare". It sounds to me like this kid really doesn't see her mom very much, but sees your daughter all the time on a daily basis.
They are only 8 years old, after all, not 18. Deciding she wanted to go home and be with her mom, considering all the time she had probably already spent with your daughter all week, is not all that unusual for an 8 year old. I'm sorry your daughter's feelings were hurt, but you should really be encouraging her to find more friends.
I used to hang out alot with a girl named Regina when I was about 8 or 9. We spent the night with each other all the time. One night I was spending the night at Regina's house and after we went to bed, I couldn't sleep and I became extremely homesick. I called my Dad to come and get me and we left while everyone was asleep. Since they lived in the country it scared the crap out of them. They thought maybe I was sleepwalking or something and fell in their lake. Her mom called my mom and they talked about it, but Regina's mom didn't go off on my mother or anything. They both understood that we were little girls. When they asked me why I was so homesick, I couldn't give them an explanation; just that I wanted to sleep in my own bed at my own house.
None of that ever stopped Regina and I from being friends, nor our mothers. We were right back at each other's houses in a day or too.
I guess my point is, don't make a huge mountain out of a molehill. It's not the only birthday party your daughter will ever have. They are only 8 and 9 years old.